chapter 8 and 9 lectures run together a little krönig-penney model and the origin of bands and band...
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Chapter 8 and 9 lectures run
together a little
Krönig-Penney Model and the
Origin of Bands and Band Gaps
For review of Schrodinger equation: http://web.monroecc.edu/manila/webfiles/spiral/6schrodingereqn.pdf
Project topics due today! Next HW due in one week. Test corrections due in 9 days.
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Learning Objectives for Today
After today’s class you should be able to: Explain the meaning and origin of energy
bands and “forbidden band gaps”Finally understand difference between
metals, semiconductors and insulators!
(If time) Relate DOS to energy bands
Another source on today’s topics, see Ch. 7 of Kittel or search Kronig-Penney model
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Using Bloch’s Theorem: The Krönig-Penney Model
Bloch’s theorem allows us to calculate the energy bands of electrons in a crystal if we know the potential energy function.
Each atom is represented by a finite square well of width a and depth V0. The atomic spacing is a+b.
We can solve the SE in each region of space: ExV
dx
d
m
)(
2 2
22
0 < x < aiKxiKx
I BeAex )( m
KE
2
22
-b < x < 0
V
x0 a a+b
2a+b 2(a+b)
V0
-b
xxII DeCex )(
mEV
2
22
0
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Boundary Conditions and Bloch’s Theorem
x = 0
The solutions of the SE require that the wavefunction and its derivative be continuous across the potential boundaries. Thus, at the two boundaries (which are infinitely repeated):
iKxiKxI BeAex )(
xxII DeCex
)(
Now using Bloch’s theorem for a periodic potential with period a+b:
x = a )(aBeAe IIiKaiKa
DCBA (1) )()( DCBAiK (2)
)()()( baikIIII eba k = Bloch
wavevector
Now we can write the boundary conditions at x = a:
)()( baikbbiKaiKa eDeCeBeAe (3)
)())()(()()( baikbbiKaiKa eDeikCeikBeikiKAeikiK (4)
The four simultaneous equations (1-4) can be written compactly in matrix form Let’s start it!
)()( xeRx ikR
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Results of the Krönig-Penney Model
Since the values of a and b are inputs to the model, and depends on V0 and the energy E, we can solve this system of equations to find the energy E at any specified value of the Bloch wavevector k. What is the easiest way to do this?
0
)()()()(
1111
)()(
)()(
D
C
B
A
eeikeeikeikiKeikiK
eeeeee
iKiK
baikbbaikbiKaiKa
baikbbaikbiKaiKa
Taking the determinant, setting it equal to zero and lots of algebra gives:
)(coscoshcossinhsin2
22
bakbKabKaK
K
By reducing the barrier width b (small b), this can be simplified to:
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Graphical Approach
Right hand side cannot exceed 1, so values exceeding will mean that there is no wavelike solutions of the Schrodinger eq. (forbidden band gap)
)cos(cossin2
2
kaKaKaK
b
Ka
Plotting left side of equation
Problems occur at Ka=N or K=N/a
)(coscoshcossinhsin2
22
bakbKabKaK
K
small b
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Turning the last graph on
it’s side )cos(cossin2
2
kaKaKaK
b
ka/
En
erg
y in
term
s o
f E
0
2
22
0 2maE
This equation determines the energy bands.
For values of K where the left side of the equation has a magnitude < 1, then k is real and energy bands are
allowed.
BAND 1
BAND 2
Forbidden band gap
m
KE
2
22
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Greek Theater Analogy: Energy Gaps
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Energy Levels of Single vs Multiple Atoms
Single Atom
Multiple Atoms
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10
Ideal Double Quantum Wells
How do we
start?
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The two solutions have different energies
Symmetric (Bonding) and Antisymmetric (Antibonding)
http://www.personal.leeds.ac.uk/~eenph/QWWAD/
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Energy vs. Barrier Width
What would 3 wells look like?
Spins not coupled
What happens as make b go to 0?
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13
Triple Quantum WellsWhich has the lowest energy?
Any relation between nodes and energy?
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14
Quadruple Quantum Wells
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15
Five Quantum Wells
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Figure 1.7: Coupled Well Energies
How would the energy levels
look for multiple wells?
What happens to these levels as the atoms get closer (b smaller)?
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Band Overlap
Often the higher energy bands become so wide that they overlap with the lower bands
Many materials are conductors (metals) due to the “band overlap” phenomenon
Also partly allows hybridization, like in carbon
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18
Energy Band Overlap
14Si: 3s23p2 Out of 8 possible n=3 electrons (2s and 6p)
Valence BandTypically the last
filled energy band
Conduction BandThe bottom
unfilled energy band
Mixing of bands known as hybridization (Si=sp3)
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19
Energy Band Formation
Valence Bandlast filled
Conduction Bandbottom unfilled
MetalNo gap
SemiconductorSmall gap (<~1eV)
InsulatorBig gap (>~1eV)
Diagram (flat or with momentum k) showing energy levels is a band diagram.
This is at T=0. What happens at higher T?
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Semiconductor Flat Band Diagram
(Quantum Well)
In
What do I mean by flat? 1. Before any movement of charge, could cause bands to bend2. At a single point in the crystal (changes with momentum)
1.43 eV
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What happens as you approach
the gap? )cos(cossin2
2
kaKaKaK
b
ka/
En
erg
y in
term
s o
f E
0
2
22
0 2maE
Classically E = ½ m v2
What happens to v as k gets close to
Brillouin zone edge?
BAND 1
BAND 2
Forbidden band gap
m
KE
2
22
)(1
knk
v
Find v for the free electron energy.
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Compare to the free-electron model
Free electron dispersion E
k
Let’s slowly turn on the periodic potential
–/a /a
22 2 2( )
2 x y zE k k km
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Electron Wavefunctions in a Periodic Potential
(Another way to understand the energy gap)
Consider the following cases:
Electrons wavelengths much larger than atomic spacing a, so wavefunctions and energy bands are nearly the same as above
01 V)( tkxiAe
m
kE
2
22
ak
V
01
Wavefunctions are plane waves and energy bands are parabolic:
E
k–/a /a
V
x0 a a+b
2a+b 2(a+b)
V1
-b
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Wavelength much greater than atomic spacing
Similar to how radio waves pass through us without affecting
Energy of wave
What happens as I lower this energy?
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Electron Wavefunctions in a Periodic PotentialU=barrier potential
Consider the following cases:
Electrons wavelengths much larger than a, so wavefunctions and energy bands are nearly the same as above
01 V)( tkxiAe
m
kE
2
22
ak
V
01
Wavefunctions are plane waves and energy bands are parabolic:
ak
V01 Electrons waves are strongly back-scattered (Bragg
scattering) so standing waves are formed:
tiikxikxtkxitkxi eeeAeeC
21)()(
ak
V01 Electrons wavelengths approach a, so waves begin to
be strongly back-scattered by the potential:)()( tkxitkxi BeAe
AB
E
k–/a /a
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The nearly-free-electron model (Standing Waves)
Either: Nodes at ions
Or: Nodes midway between ions
a
Due to the ±, there are two such standing waves possible:
titiikxikx ekxAeeeA )cos(2
21
21
titiikxikx ekxiAeeeA )sin(2
21
21
These two approximate solutions to the S. E. at have very different potential energies. has its peaks at x = a, 2a, 3a, …at the positions of the atoms, where V is at its minimum (low energy wavefunction). The other solution, has its peaks at x = a/2, 3a/2, 5a/2,… at positions in between atoms, where V is at its maximum (high energy wavefunction).
ak
tiikxikx eeeA
21
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The nearly-free-electron model
Strictly speaking we should have looked at the probabilities before coming to this conclusion:
a
~ 2
2
2
titiikxikx ekxAeeeA )cos(2
21
21
titiikxikx ekxiAeeeA )sin(2
21
21
)(cos2 22*axA
)(sin2 22*axA
Different energies for electron standing waves
Symmetric and
Antisymmetric Solutions
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28
E
k
Summary: The nearly-free-electron model
BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE
-2π/a –π/a π/a 2π/a
In between the two energies there are no allowed energies; i.e., wavelike solutions of the Schrodinger equation do not exist.
Forbidden energy bands form called band gaps.
The periodic potential V(x) splits the free-
electron E(k) into “energy bands” separated by
gaps at each BZ boundary.
E-
E+
Eg
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E
k
Approximating the Band Gap
BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE
-2π/a –π/a π/a 2π/a
a
xax
a dxxVEE0
22 )(cos)(
E-
E+
Eg
a
x
g dxxVEEE0
22])[(
For square potential: V(x) =Vo for specific values of x (changes integration limits)
)(cos2 22*axA
)(sin2 22*axA